A note on the generalized reflexion of Guitart and Lair

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

G. M. K ELLY

A note on the generalized reflexion of Guitart and Lair

Cahiers de topologie et géométrie différentielle catégoriques, tome 24, n

^o

2 (1983), p. 155-159

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A NOTE ON THE GENERALIZED REFLEXION OF GUITART AND LAIR

by G. M. KELLY*

CAHIERS DE TOPOLOGIE ET

GÉOMÉTRIE DIFFÉRENTIELLE

Vol. XXIV - 2

(1983)

By

^{a weak}

reflexion

^{of a}

locally-small category (i

^{onto a full sub-}

category

^{we mean the}

assigning

^to

each A 6 D

of a small

projective

cone rA, ^{with vertex A} and with base

in B,

^such

that D(rA, ^{B )}

^{is a}

colimit-cone in

Set

for each

A f S

and each B E

B.

When each rA ^has its base indexed

by

^{a discrete} category, ^{77 is a}

multi-reflexion

^{in the sense}

of Diers

[ 1 );

^{it is an} actual reflexion if moreover each of these discrete

categories

^{is 1 .}

For

example,

^let

d

be the category of commutative

rings. When B

consists of local

rings,

^a weak reflexion is

given by taking

^for 7TA ^the

cone of localizations A-&#x3E;

Ap

^of

^{A ;}

its base is indexed

by

^{the ordered set}

of

prime

^{ideals p of A .}

When 93

consists of the

fields,

^a multi-reflexion is

given by

^the discrete cone A 4

A/m

where m runs

through

the maximal ideals of A .

When fl

consists of the

rings

^{A with 2 A}

= 0 ,

^{an actual re-}

flexion is

given by A - ^{A/ 2A .}

Guitart and Lair

study

ⁱⁿ

[4]

the existence of weak reflexions when

B

is

given

^as follows. We have a set e ⁼

100 ^{} o f} projective

^cones

in

8,

^where

L! N f3

denotes the

functor

constant at

N F3

^;

and B consists o f

those A E

Q for

^{whi ch each}

D (0B, A) is a

colimi t- cone ⁱⁿ

Set . They

further restrict themselves to the

special

^{case in which each}

generator o f

each cone

e (3

^{is an}

epimorphism

ⁱⁿ

(f.

Each of the

examples

above is of this kind. For local

rings

^there

are two cones

01 and ()2

ⁱⁿ

^{8 ;} 0,

^{is the}

^{pushout diagram}

^{of the two}

^(epi-

morphic)

maps

* The author

gratefully acknowledges

the assistance of the Australian Research Grants Committee.

while

02

^{is the cone of vertex 0 over the empty}

^diagram.

^{For fields there}

are

again

two cones :

e2

^{as above and}

03

^{the discrete cone}

For

rings

^{with 2 A =}

0 ,

^{there is a}

single

^cone

e¢

whose base is indexed

by 1 , namely

^{Z 7}

^{Z/ 2Z.}

We suppose henceforth

that 93

is

given

^as above. We recall

that,

for a

regular

^cardinal a, ^an

object

^A

E D

is called

a-presentable

^if

D(A, -) : D-&#x3E; Set

_preserves

a -filtered

colimits. Guitart and Lair sketch in

[4] ]

^{a rather}

complicated proof by

transfinite induction of the

following:

There is a weak

reflexion

⁷⁷

o f (t onto 93 if ⁽ⁱ

^is

cocomplete, if

^each

L B

is

small,

and

if

^{there is a}

regular

^{cardinal a such that each}

N 18

^{and each}

T B L

^{is a}

-presentabl e.

^{Moreover rr can be taken to be a}

mul ti-re fl exion

if each LB

^{is discrete.}

The a

-presentability hypothesis

^{is a} strong one ;

hardly

any ob-

jects

^are

a-presentable

in the category of

topological

spaces ^or in the dual of an

algebraic

category.

By analogy

^{with the case where}

each f p

^{is 1}

- the

«orthogonal subcategory problems

^of

[2] -

^this

hypothesis

^{should not}

be needed when the generators ^{of the cones}

Op

^are

epimorphic :

^{at least}

if

Q

is

cowellpowered,

^{which is not a} grave restriction.

By

^{the same ana-}

logy,

^{there should be a}

simple

and direct

proof

^{in this case. We now}

verify

that this is _so, and that moreover the base of each cone rA may then be taken to be an ordered set.

We refer to

[5] ]

for the notion of

strong monomorphism,

^{and for the}

fact that

epimorphisms

and strong

monomorphisms

constitute a factorization system

(see [2]) ^{on d} if (t

admits finite limits and all intersections of strong

monomorphisms,

^{or if}

8

admits finite colimits and all cointersec- tions of

epimorphisms ; certainly, therefore,

^if

d

is

complete

^{and well-}

powered,

^or

cocomplete

^and

cowellpowered.

THEOREM 1. L et the

full subcategory 93 of

^the

locally-small category D

be determined as above

by

^{a set}

^{fl (not} necessarily ^small) of

^cones

0B

(not necessarily ^small),

where each

generator o f

^each

0B

^is

^epimorphic

in

(t.

Let

epimorpbisms

^and

strong monomorphisms

constitute ^a

factoriza-

tion system ^on

(i,

and let

(t

be

cowellpowered.

For each A E

(i

denote

by SA the

^small

^category

^whose

^objects

^are

(a

^set

o f representatives o f)

^the

epimorphisms p :

A -&#x3E; C in

(T

with domain A and codomain in

B,

^{and whose}

maps p -&#x3E; p’

^{are the}

maps

q : C -&#x3E; C’

with

qp= p’ ; clearly SA

^{is an ordered set. Let}

dA : SA -&#x3E;B ^D

^{be the}

projection functor sending ^{p :}

A -&#x3E; C to C, and let

be the cone whose

p-th component

^is

p itself.

Then an

object

^B

of (i

^lies

in fl if

^and

only if

^each

Cl(

_7TA,

B )

is a colimit-cone in

Set.

PROOF. The essential observation is

that

is closed

in Sunder

_strong

subobjects.

^{To see this it suffices to consider a}

single

^{cone e :}

AN-&#x3E;

T of

0 ,

^with

epimorphic

generators

0i :

^{N 4}

Ti . ^{Let j :}

^{D 4 B be a strong}

monomorphism

ⁱⁿ

S,

^with

B E B. By

^the

diagonal-fill-in

property ^for

epi- morphisms

^and strong

monomorphisms,

^the

diagram

is a

pullback

ⁱⁿ

^{Set .}

Since colimits are universal in

Set ,

^{and since}

D (0i, ^{B )}

^{is a} colimit-cone in

Set ,

^{so is}

D(0i, ^{D ) ;}

^{so that D f}

^J3.

It is now easy ^{to see}

that D(rA, ^{B )}

^{is a} colimit-cone for

B f 93 .

For let

f :

^{A 4} B, and ^let

f

^{factorize as an}

epimorphism p: A 4

^{C follow-}

ed

by

^a strong

monomorphism j :

C-&#x3E; B. Since

C f 93 _by

the

above, p

^{is a} generator

of uA through

^which

f

factorizes. If

f

^also factorizes as g

p’

through

^another generator

p’: A 4

^C’

of rA,

^the

diagonal-fill-in

property

applied to g p’ = j p gives a q:

C’4 C with

qp’=P and j q = g .

^Hence

D(rA, ^{B )}

^{is a} colimit-cone.

Conversely,

^if

D(rA, B)

^{is a} colimit-cone for each

A ,

^then

D(rB, ^{B )}

^{is a}

colimit-corie ;

^{so that 1 :} B-&#x3E; B factorizes ^as

1=jp

^for

some

epimorphism p :

B-&#x3E; C with C f

93.

But then the

epimorphism p ,

^be-

ing

^a

coretraction,

^is

invertible ;

^{and B c}

93.

D

THEOREM 2. Add to the

hypotheses of

^{Theorem 1 the}

completeness of D,

and

suppose

^{each cone}

0B

^{to have a discrete}

base 26.

Then the restric- tion

of 17 A

^{to a suitable}

full subcategory of SA ^gives

^a

multi-re flexion of a

^onto

93.

P ROO F. Since connected limits commute with discrete colimits in

Set,

we

have 93

closed in

8

under connected limits. For each connected com-

ponent

6

of

SA , therefore,

the limit of

dA|d : d-&#x3E;SA -&#x3E; ^Q

^{is an}

^object Eo. of 93;

^and

the p.-A4C

^of

SA

^{induce a map}

r s :

A-&#x3E;

E8 .

^{Let this}

factorize as the

epimorphism sd : A -&#x3E; K3

^followed

^by

the strong ^mono-

morphism kd : K,5 4 Ed.

^Then

K 8

^-:

^.93,

^and

^sd

^{is an}

^object

^of

SA ;

^clear-

ly

^the greatest

object

^{of the ordered set}

SA

^which

^belongs

^{to 5 . It is now}

evident _{that any}

f :

A -&#x3E; B with

B =- 93

factorizes

uniquely through

^some

sd,

^and

through

^one

only.

⁰

We include for

completeness

the classical :

THEOREM

3. If each 2 o

^{= 1 in Theorem}

^{2, B}

^{is closed under limits in}

(i,

^{and we}

get

^{an actual}

re flexion _{10 A} o f (i

^onto

B,

^where

_{p A}

^{is the}

epi- morphic part of

^the

factorization of

A -&#x3E; lim

d A

^{into an}

epimorpbism fol-

lowed

by

^a

strong monomorphism.

^CI

We end

by observing

^{that the}

cowellpoweredness hypothesis

^of

Theorem 1 does hold in the

example

^{to which Guitart and Lair}

give

^most

prominence -

that of the

algebras

^{for a mixed sketch S .}

By

^{this is meant}

a small category

8

in which are

given

^a small set (D =

{O a ^I

^{of small pro-}

jective

^{cones and a} small set T

= {UB}

^{of small inductive} cones; unlike Guitart and

Lair,

^{we do not ask}

the 0.

^{to be} limit-cones nor the

UB

^{to be}

colimit-cones. The category

S-Alg

^of

S-algebras

^{is the full}

subcategory

of

[§,Set] given by

those A :

S-&#x3E; Set

for which each

A Oa

^{is a limit-}

cone and each

AYB

^{is a} colimit-cone. The sketch S is

projective

^when

the

set IF

is empty ; ^write

So

^{for the}

projective

^sketch obtained from S

by discarding ^Y

^{It is classical}

that categories

^{of the form}

So-Alg

^are

the

locally presentable

^ones of Gabriel-Ulmer

[3] ;

^and that such a cat-

egory is reflective in

[S, Set],

^{and is therefore}

complete

^and

cocomplete.

Let ^{Z :}

SOP So -Alg

^{be the}

composite

^{of the Yoneda}

embedding

Y :

S op-&#x3E; Set ]

and the reflexion R :

Set] , S o -A lg . Clearly B =

S-Alg

^{is the full}

subcategory ^{of 8}

⁼

So-Alg consisting

^{of those}

objects

A such that

8f-, A )

^{sends the}

projective

^cone

0 B = ZYB ^{of lt}

^{to a co-}

limit-cone in

Set

for each

f3.

Note that each generator ^of

0B

^is

^epimor-

phic

^{if each} generator ^of

zP f3

^is

monomorphic.

Finally,

observe that

S

is

cowellpowered by

Satz 7.14 of

[3],

^an

account of which in

English

^can be found in Section

8.6

of

[6].

BIBLIOGRAPHY.

1. Y. DIERS, ^Families universelles de

morphismes,

^{Ann. Soc.} Sci. Bruxelles 93 (1979), 175-195.

2.

P. J.

^FREYD &#x26; G. M. KELLY,

Categories

^of continuous functors I, J. Pure

Appl.

Algebra

2 (1972), 169- 191; ^Erratum 4 (1974), ^121.

3. P. GABRIEL &#x26; F. ULMER, ^Lokal

prasentierbare Kategorien,

Lecture Notes in Math. 221,

Springer-Verlag,

^1971.

4. R. GUITART &#x26; C. L AIR, Existence de

diagrammes

localement

libres, Diagram-

me s

6 ( 198 1 ).

5. G. M. KELLY,

Monomorphisms, epimorphisms,

^and

pull-backs,

J. Austral. Math.

Soc. 9

(1969),

^{124- 142.}

6. G. M. KELLY, Structures defined

by

finite limits in the enriched context I, ^Ca- hiers

Top.

^{et Géom.}

Diff.

^XXIII-

1 ( 1982),

^3-42.

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